132 ANALYTICAL MECHANICS If we plot the last equation for the four different cases of the initial velocity we obtain the curves of Fig. 70. It is evident from these curves that a very short time after the beginning of the motion the distance covered increases at a constant rate, as would be expected from the meaning of the limiting velocity. CASE II. RESISTANCE PBOPOR- TIONAL TO THE SQUARE OF THE VELOCITY. — The assumption that resistance varies as the velocity holds only for slowly moving bodies. It is found that for projectiles whose velocities lie under 1000 feet per second and over 1500 feet per second the resistance varies, approximately, as the square of the velocity, while between these values it varies as the cube and even higher powers of the velocity. The experimental data on the subject are not enough to find a law of variation which holds in all cases. If we assume the resistance to vary as the square of the velocity, then the force equation for a falling body becomes FIG. 70. or dv -r dv • (5) where k = — = constant. In order to integrate the last equation we replace ^ by v~=- and rearrange the terms so that we dt get v dv = — k ds.